Chapter 2: Iterative Combinatorial Auctions

Transcription

1 Chapter 2: Iterative Combinatorial Auctions David C. Parkes 1 Introduction Combinatorial auctions allow bidders to express complex valuations on bundles of items, and have been proposed in settings as diverse as the allocation of floor space in a new condominium building to individual units (Wired 2000) and the allocation of take-off and landing slots at airports (Smith, Forward). Many applications are described in Part V of this book. The promise of combinatorial auctions (CAs) is that they can allow bidders to better express their private information about preferences for different outcomes and thus enhance competition and market efficiency. Much effort has been spent on developing algorithms for the hard problem of winner determination once bids have been received (Sandholm, Chapter 14). Yet, preference elicitation has emerged as perhaps the key bottleneck in the real-world deployment of combinatorial auctions. Advanced clearing algorithms are worthless if one cannot simplify the bidding problem facing bidders. Preference elicitation is a problem both because of the communication cost of sending bids to the auction and also because of the cost on bidders to determine their valuations for different bundles. The problem of communication complexity can be addressed through the design of careful bidding languages, that provide expressive but concise bids (Nisan Chapter 9). Non-computational approaches can also be useful, such as defining the good and bundle space in the right way in the first place (Pekeč and Rothkopf Chapter 16). However, even well-designed sealed-bid auctions cannot address the problem of hard valuation problems because they preclude the use of feedback and price 1

2 discovery to focus bidder attention. There are an exponential number of bundles to value in CAs. Moreover, the problem of valuing even a single bundle can be difficult in many applications of CA technology. For instance, in the airport landing slot scenario (see Ball, Donohue and Hoffman Chapter 20) we should imagine that airlines are solving local scheduling, marketing, and revenue-management problems to determine their values for different combinations of slots. Iterative combinatorial auctions are designed to address the problem of costly preference elicitation that arises due to hard valuation problems. An iterative CA allows bidders to submit multiple bids during an auction and provides information feedback to support adaptive and focused elicitation. For example, an ascending price auction maintains ask prices and allows bidders to revise bids as prices are discovered. Significantly, it is often possible to determine an efficient allocation without bidders reporting, or even determining, exact values for all bundles. In contrast, any efficient sealed-bid auction requires bidders to report and determine their value for all feasible bundles of goods. This ability to mitigate the preference elicitation problem is a central concern in iterative CA design. But there are also a number of less tangible yet still important benefits: Iterative CAs can help to distribute the computation in an auction across bidders through the interactive involvement of bidders in guiding the dynamics of the auction. Some formal models show the equivalence between iterative CAs and decentralized optimization algorithms (Parkes and Ungar 2000a, de Vries, Schummer, and Vohra 2003). Iterative CAs can address concerns about privacy because bidders only need to reveal partial and indirect information about their valuations. 1 Transparency is another practical concern in CAs. In the high-stakes world of wireless spectrum auctions, the Federal Communications Commission 2

3 (FCC) has been especially keen to ensure that bidders can verify and validate the outcome of an auction. Although mathematically elegant, the VCG outcome can be difficult to explain to bidders, and validation requires the disclosure and verification of many bids, both losing and winning. Thus, even as readily describable implementations of sealed-bid auctions, iterative CAs can offer some appeal (Ausubel and Milgrom 2002). The dynamic exchange of value information between bidders, that is enabled within iterative CAs, is known to enhance revenue and efficiency in single item auctions with correlated values (Milgrom and Weber 1982). Although little is known about the design of iterative CAs for correlated value problems, one should expect iterative CAs to retain this benefit over their sealed-bid counterparts. Certainly, correlated value settings exist: consider the wireless spectrum auctions in which valuations are in part driven by underlying population demographics and shared technological realities. Yet, even with all these potential advantages iterative CAs offer new opportunities to bidders for manipulation. The biggest challenge in iterative CA design is to support incremental and focused bidding without allowing new strategic behavior to compromise the economic goals of efficiency or optimality. For instance, one useful design paradigm seeks to implement auctions in which straightforward bidding (truthful demand revelation in response to prices) is an ex post equilibrium. This equilibrium is invariant to the private information of bidders, so that straightforward bidding is a best response whatever the valuations of other bidders. Steps can also be taken to minimize opportunities for signaling through careful control of the information that can be shared between bidders during an auction. Finally, the benefits of iterative auctions disappear when bidders choose to strategically delay bidding activity until the last rounds of an auction. Activity 3

4 rules (Milgrom 2000) can be used to address this stalling and promote meaningful bidding during the early rounds of an auction. The existing literature on iterative CAs largely focuses on the design of efficient auctions. Indeed, there are no known optimal (i.e. revenue-maximizing) general-purpose CAs, iterative or otherwise. As such, the canonical VCG mechanism (see Chapter 1) has guided the design of many iterative auctions. 2 We focus mainly on price-based approaches, in which the auctioneer provides ask prices to coordinate the bidding process. We also consider alternative paradigms, including decentralized protocols, proxied auctions in which a bidding agent elicits preference information and automatically bids using a predetermined procedure, and direct-elicitation approaches. In outline, Section 2 defines competitive equilibrium (CE) prices for CAs, which may be non-linear and non-anonymous in general. Connections between CE prices, the core of the coalitional game, and the VCG outcome are explained. Section 3 describes the design space of iterative CAs. Section 4 discusses price-based auctions, providing a survey of existing price-based CAs in the literature and a detailed case study of an efficient ascending price auction. Section 5 considers some alternatives to price-based design. Section 6 closes with a brief discussion of some of the open problems in the design of iterative combinatorial auctions, and draws some connections with the rest of this book. 2 Preliminaries Let G = {1,..., m} denote the set of items, and assume a private values model with v i (S) 0 to denote the value of bidder i I = {1,...,n} for bundle S G. Note that set I does not include the seller. We assume free-disposal, with v i (T) v i (S) for all T S, and normalization, with v i ( ) = 0. Let V denote the set of bidder valuations. Bidders are assumed to have quasi-linear utility (we also use payoff interchangeably with utility), with utility u i (S,p) = v i (S) p for bundle S at price p 0. This assumes the absence of any budget constraints. 4

5 Further assume that the seller has no intrinsic value for the items. The efficient combinatorial allocation problem (CAP) solves: max v i (S i ) S=(S 1,...,S n) i I s.t. S i S j =, i,j [CAP(I)] Let S denote the efficient allocation. Also, we write CAP(I \ i) to denote the combinatorial allocation problem without bidder i. 2.1 Competitive Equilibrium Prices We can consider a hierarchical structure for ask prices in CAs: Linear. Prices p j 0, for j G, define additive prices on bundles, with p(s) = j S p j. Non-linear. Prices, p(s) 0, for S G, allow p(s) p(s 1 ) + p(s 2 ), for some S = S 1 S 2 and S 1 S 2 =. Non-linear and Non-anonymous. Prices p i (S) 0, allow discriminatory pricing, with p i (S) p i (S) for bidder i i, in addition to non-linear prices. In the following definitions we adopt p i (S) for notational convenience. We intend to allow (but not require) with this notation non-linear and non-anonymous prices. For instance, linear prices p j can be considered to induce prices p i (S) = j S p j for bundle S and bidder i. Competitive equilibrium prices extend the concept of Walrasian equilibrium prices to a CA. Let π i (S,p) = v i (S) p i (S) denote bidder i s payoff from bundle S at prices p and Π s (S,p) = i I p i(s i ) denote the seller s revenue from allocation S at prices p. 5

6 Definition 1 (Competitive Equilibrium). Prices, p, and allocation S = (S 1,...,S n) are in competitive equilibrium (CE) if: π i (Si,p) = max [v i(s) p i (S),0] i (1) S G Π s (S,p) = max p i (S i ) (2) S Γ i I where Γ denotes the set of all feasible allocations. A competitive equilibrium (p,s ) is such that allocation S maximizes the payoff of every bidder and the seller given prices. Allocation S is said to be supported by prices p in CE. Theorem 1. Allocation S is supported in competitive equilibrium if and only if S is an efficient allocation. This welfare theorem follows from a simple linear-programming (LP) duality argument for suitably extended LP formulations of the CAP (Bikhchandani and Ostroy 2002, also Chapter 8). Moreover, CE prices always exist for the CAP. For instance, prices p i = v i trivially satisfy the CE conditions. The main new element in CAs is that these CE prices must sometimes be non-linear and non-anonymous. Bikhchandani and Ostroy also show an equivalence between the core of the coalitional game and the set of CE prices. All core outcomes can be priced, and all CE prices correspond to core payoffs. Many iterative CAs are designed to converge to CE prices, and as such it is important to characterize classes of valuations for which linear, and non-linear but anonymous, CE prices exist. We will also see that it is necessary that an efficient CA must determine enough information about bidder valuations to define a set of CE prices, and necessary that a Vickrey auction determines enough information to define a set of universal CE prices. For the existence of linear CE prices, it is sufficient (and almost necessary) 3 that valuations satisfy a goods are substitutes property (Kelso and Crawford 1982, Gul 6

7 and Stacchetti 1999). This substitutes condition is defined indirectly, with respect to a demand set: D i (p) = {S : π i (S,p) max T G π i(t,p),π i (S,p) 0,S G}, (3) which includes all bundles that maximize a bidder s payoff at the prices. Definition 2 (Goods are Substitutes). Valuation v i satisfies goods are substitutes if for all linear prices p,p such that p p (component-wise), and all S D i (p), there exists T D i (p ) such that {j S : p j = p j } T. The goods are substitutes (or simply substitutes) condition requires that a bidder will continue to demand items that do not change in price as the price on other items increases. Substitutes valuations include unit-demand valuations with v i (S) = max j S {v ij } for all S and value v ij on item j in isolation, but preclude the possibility of items with complementary values (Lehmann, Lehmann, and Nisan 2001). Conditions for the existence of non-linear but anonymous CE prices are less well-understood, but sufficient conditions presented in Parkes (2001) (Theorem 4.7) include supermodular valuations, single-minded bidders that value a particular bundle, and bidders with safe valuations such that each pair of bundles with positive value to a bidder share at least one item. Consider, for example, a bidder in the FCC spectrum auction that definitely needs lower Manhattan, along with as many of the geographically neighboring licenses as possible. 2.2 Minimal Competitive Equilibrium Prices In fact, many iterative CAs are designed to converge to minimal CE prices. This can be useful for two reasons. First, minimal CE prices on bundles in the efficient allocation correspond to VCG payments for a restricted class of valuations. In this case, we say that the CE prices support the VCG payments. Termination with CE prices that support VCG payments brings straightforward bidding into an ex post equilibrium. Second, Ausubel and Milgrom (2000, also Chapter 3) show that 7

8 implementing minimal CE prices (corresponding to buyer-optimal core outcomes) avoids the problems that can occur with the VCG auction when VCG payments are not supported with minimal CE prices. Definition 3 (Minimal CE Prices). Minimal CE prices minimize the seller s total revenue Π s (S,p) on the efficient allocation S across all CE prices. A bidder s payment in the VCG mechanism is always less than or equal to the payment by that bidder at any CE price (Bikhchandani and Ostroy 2002). Thus, minimal CE prices always provide an upper-bound on VCG payments. Moreover, a bidder s VCG payment is equal to the CE price on her efficient bundle in some CE (Parkes and Ungar 2000b). A characterization in terms of the coalitional value function explains when the VCG can be supported simultaneously to all bidders in the minimal CE. Let w(l) for L I denote the coalitional value for a subset L of bidders, equal to the value of the efficient allocation for CAP(L). The buyers are substitutes (BAS) condition requires, w(i) w(i \ K) [w(i) w(i \ i)], K I (BAS) i K Theorem 2. (Bikhchandani and Ostroy 2002) A buyers are substitutes (BAS) coalitional value function is necessary and sufficient to support the VCG payments in competitive equilibrium. In particular, the VCG payments are implemented in the minimal CE (or buyer-optimal core) when BAS holds, and buyer-optimal core payoffs are unique exactly when BAS holds. A number of ascending price CAs can only terminate with minimal CE prices given a slightly stronger condition, that of a buyer-submodular (BSM) coalitional value function: w(l) w(l \ K) [w(l) w(l \ i)], K L, L I (BSM) i K 8

9 Bikhchandani and Ostroy (Chapter 8) refer to BSM as buyers are strong substitutes. Clearly, a BSM coalitional value function also satisfies BAS. But there are cases for which values satisfy BAS but not BSM (see Ausubel and Milgrom 2002, Section 7, for example). Interestingly, substitutes valuations implies BSM and is almost necessary. Roughly, if at least one bidder does not satisfy substitutes then one can construct substitutes valuations for other bidders such that the coalitional value function fails BSM. See Ausubel and Milgrom (Chapter 1) for further discussion. Thus, the same conditions for the existence of a linear price equilibrium are sufficient and almost necessary for the existence of some price equilibrium (although perhaps non-linear and non-anonymous) that supports the Vickrey outcome Universal Competitive Equilibrium Prices Experiments have suggested that BAS can often fail in realistic settings for CAs. 5 In these cases the VCG payments are not supported in any price equilibrium. We can still design price-based CAs by characterizing a stronger condition on CE prices that implies enough information to determine VCG payments from these prices. For this, we restrict attention to the universal CE prices (Parkes and Ungar 2002, Mishra and Parkes 2004). Definition 4 (Universal CE Prices). Prices p are universal Competitive Equilibrium (UCE) prices if: a) Prices p are CE prices. b) Prices p i are CE prices for CAP(I \ i), meaning they support some efficient allocation in CAP(I \ i), for all bidders i. where p i = (p 1,...,p i 1,p i+1,...,p n ). In words, prices are UCE when an efficient allocation for the restricted allocation problem without bidder i is supported with prices p i, for each bidder i removed 9

10 in turn. Thus, UCE prices are CE prices in the main economy and in every marginal economy. Note that UCE prices need not require that the same allocation is supported in every marginal economy. The prices must support some efficient allocation in each marginal economy. 6 UCE prices always exist, for example p i = v i, for all bidders i, are UCE prices. Moreover, a universal price equilibrium provides sufficient information about bidder valuations to compute the VCG outcome. Theorem 3. (Parkes and Ungar 2002) Given a UCE with prices p uce and an efficient allocation S, the VCG payment to bidder i is computed as: p vcg,i = p uce,i (S i ) [Π I(p uce ) Π I\i (p uce)] (4) where Π L (p) = max S Γ i L p i(s i ) for bidders L I. In the special case when prices are equal to valuations then this adjustment is equivalent to the standard definition of VCG payments. 2.4 Informational Requirements Both CE and UCE prices have a central role in the preference elicitation problem. First, any auction that implements an efficient allocation must determine a set of CE prices. Second, any auction that implements the Vickrey outcome must determine a set of UCE prices. Segal (Chapter 11) provides an extended discussion. Since the VCG auction is basically unique amongst the class of efficient auctions that take a zero payment from losing bidders (Ausubel and Milgrom, Chapter 1), these equivalences confirm the central role of prices in developing iterative CAs. Theorem 4. (Parkes 2002, Nisan and Segal 2003) A combinatorial auction realizes the efficient allocation if and only if the auction also realizes a set of CE prices and an allocation supported in the price equilibrium. 10

11 This result requires a technical condition of privacy-preservation, which precludes bidders from making their valuations contingent on the valuations of other bidders (e.g. my value for A is at least bidder 2 s value for A ). 7 Theorem 5. (Parkes and Ungar 2002, Lahaie and Parkes 2004b) A combinatorial auction realizes the VCG outcome if and only if the auction also realizes a set of UCE prices and an allocation supported in the price equilibrium of the main economy. That UCE prices provide sufficient information was first proved in Parkes and Ungar (2002). The necessary direction is due to Lahaie and Parkes (2004b). It is important to realize that the CE and UCE prices referenced in these results may only be realized implicitly and are not necessarily explicitly constructed in the auctions. Considering minimal CE prices in particular, Mishra and Parkes (2004) note that minimal CE prices are universal iff BAS holds. In general, UCE prices are greater than the minimal CE prices because they must consider competition in the marginal economies in addition to the main economy. The informational equivalence between the efficient outcome and the problem of discovering CE prices leads to a (largely negative) characterization of the worst-case communication complexity and preference-elicitation requirements of any efficient CA, iterative or otherwise (Segal, Chapter 11). On the other hand, iterative CAs are designed to have good elicitation properties on typical instances, while sealed-bid auctions must suffer the worst case every time. Moreover, this price equivalence suggests the central role of prices in the design of iterative CAs. Any protocol to determine the VCG outcome must (implicitly) determine UCE prices, so why not construct protocols to converge directly to UCE prices? We return to this theme in Section 4. 11

12 2.5 Examples The following examples illustrate the concept of CE and UCE prices and also serve to illustrate the principle that it is often unnecessary to receive complete information about bidder valuations to determine the Vickrey outcome. For each example, we define a space of valuations (that contain the true valuations) that provides sufficient information to determine the Vickrey outcome. The information is minimal we call this a minimal information set in the sense that no relaxed constraints on valuations are sufficient to pin down the Vickrey outcome. Example 2.1 Consider a single-item auction with three bidders and values (10, 8, 6). The efficient allocation assigns the item to bidder 1, and the Vickrey payment is $8. Prices 10 p 8 are all in CE, and p = $8 is the unique anonymous UCE price. Notice that the UCE price must be at least $8 to satisfy CE condition (1) for bidder 2 in CAP({1,2,3}) but no greater than $8 to satisfy the same condition for bidder 2 in CAP({2,3}). The CE prices define a minimal information set, ˆV 1, defined as the subset of valuations that satisfy constraints {v 1 p,v 2 p,v 3 p,10 p 8}. UCE prices imply additional information {v 2 = 8,v 3 8}, which together with ˆV 1 is a minimal information set for the VCG outcome. Notice that an ascending price (i.e. English) auction can elicit this information if bidders 1 and 2 bid up the price to just above 8, at which point the auction terminates. Bidder 3 can remain silent. Example 2.2 Consider a combinatorial allocation problem with items {A,B} and 5 bidders (see Figure 2.1). The efficient allocation allocates A to bidder 1 and B to bidder 2 for a total value of 70. The VCG payments are p vcg,1 = 30 (70 65) = 25 and p vcg,2 = 40 (70 55) = 25. Figure 2.1 (b) illustrates an information set on 12

13 bidder valuations, that is sufficient to compute the VCG outcome and minimal in the sense that no constraints can be relaxed. The following prices are UCE for any valuation in this set: p(a) = 25,p(B) = 25,p(AB) = 25 to bidders {1,2,4,5} and prices p 3 (A) = 20,p 3 (B) = 20,p 3 (AB) = 40 to bidder 3. In fact, these prices are also minimal CE prices and the discount computed in Eq. 4 is zero for bidders 1 and 2, and BAS is satisfied (because of the presence of bidders 4 and 5). Without these bidders, the BAS condition fails and the VCG payments become p vcg,1 = 0 and p vcg,2 = 20, which can be computed from UCE prices p 1 = (20,0,20),p 2 = (0,40,40) and p 3 = (0,20,40). Additional information is needed from bidder 2 in this variation. 3 The Design Space for Iterative Combinatorial Auctions The design space for iterative CAs is larger than for one-shot auctions. Important considerations include the design of information feedback to bidders and rules to guide the submission of bids. Cramton (Chapter 4) provides an in-depth discussion of many of these issues in the design of simultaneous ascending price auctions. Let the state of an auction include all the information that is sufficient to define the future dynamics of the auction. For example, the state of an auction can define the ask prices, the provisional allocation, and also the bid improvement rules as they apply to particular bidders. Briefly, we can consider the role of the following design features: Timing issues. Iterative auctions may be continuous, allowing bids to be submitted at any time with continual updates to the current provisional allocation and prices. Alternatively, iterative auctions may be discrete, or round-based, with the state updated periodically and with bidders provided with an opportunity to revise bids between rounds. Continuous auctions can promote faster propagation of feedback information to bidders and help to quickly focus elicitation. However, 13

14 continuous combinatorial auctions can be infeasible because the winner-determination problem must be resolved whenever a new bid is submitted. Continuous auctions also lead to high monitoring and participation costs for bidders. In comparison, discrete auctions allow an auctioneer to publish a schedule for rounds in the auction and bidders can plan when to allocate time to refine their values and bids. Information feedback. Information feedback about the state of an auction can include information about the bids submitted and also aggregate information, such as price feedback and the current provisional allocation, to guide bidding. Information hiding is also possible, for example with rounding to limit the potential for signaling between bidders and with limited and discriminatory reporting of bid information. Information feedback policies make a tradeoff between serving the goal of providing effective bid guidance and minimizing the opportunity for collusion and other forms of manipulation through signaling and coordination. Bidding Rules. Ask prices are a common form of bid improvement rule, placing a lower-bound on the allowable bid price on a bundle. Bid improvement rules can also require a minimal percentage improvement over the current highest bid on a bundle, or over the total revenue in the next round given current bids. Activity rules (Milgrom 2000) introduce further restrictions, such as requiring that a bidder bids for a decreasing market share as prices increase during an auction. Ausubel, Cramton and Milgrom (Chapter 5) provide an extended discussion of bid-improvement and activity rules. Activity rules were introduced in the early FCC wireless spectrum auctions and proved important. 8 Decisions about appropriate rules are often guided by a tradeoff between providing expressiveness so that bidders can follow straightforward bidding strategies, while promoting early information 14

15 exchange between bidders and limiting the opportunity for bidders to wait and snipe at the end of an auction. Computational considerations also matter, for example linear prices can simplify the problem facing bidders in an auction (Kwasnica, Ledyard, Porter, and DeMartini 2004) but can be expensive to compute (Hoffman 2001). Termination Conditions. Auctions may close at a fixed deadline, perhaps with an opportunity for a final sealed-bid round of bidding (sometimes called a proxy round). Alternatively, auctions can have a rolling closure with the auction kept open while one or more losing bidders continue to submit competitive bids. Fixed deadlines are useful in settings in which bidders are impatient and unwilling to wait a long time for an auction to terminate. However, fixed deadlines tend to require stronger activity rules to prevent the auction reducing to a sealed-bid auction with bids delayed until the final round. In comparison, rolling closure rules have been shown to promote early and sincere bidding. 9 Bidding Languages. A bid can be a complex object and expressed in terms of logical connectives (Nisan, Chapter 9). One popular bidding language is exclusive-or (XOR), in which bid (p 1,S 1 ) xor (p 2,S 2 ) xor... xor (p l,s l ) has semantics I will buy at most one of these bundles at the stated bid price. Another popular language is additive-or (OR) bidding languages, in which bid (p 1,S 1 ) or (p 2,S 2 ) or... or (p l,s l ) has semantics I will buy one or more of these bundles at the stated bid price. Bidding languages can also place constraints on the bid prices, for example by requiring click-box bidding in which bidders must submit bids from a menu. 10 The expressiveness of a bidding language in an iterative CA must be considered together with the opportunity to refine bids during an auction. For instance, a language that is additive-or on items is not expressive in a 15

16 one-shot CA but becomes expressive in an ascending auction when bidders can decommit from bids. 11 Bidding languages are often designed to support straightforward bidding with bidders able to state the bundle that maximizes their surplus in response to prices in each round. Proxy agents. Proxy agents provide a still richer interface for iterative CAs (Parkes and Ungar 2000b, Ausubel and Milgrom 2002). Bidders can provide direct value information to an automated bidding agent that bids on their behalf within an auction. The bidder-to-proxy language should allow a bidder to express partial and incomplete information, to be refined during the auction, in order to realize the elicitation and price discovery benefits of an iterative auction. Proxy agents can query a bidder actively when they have insufficient information to submit bids. Proxy agents can also facilitate faster convergence with rapid automated proxy rounds interleaved with bidder rounds. Mandatory proxy agents can be useful in restricting the strategy space available to bidders. One concern in the design of proxy auctions is to determine when to allow proxy information to be revised and to determine the degree of consistency to enforce across revisions. An additional concern is that of trust and transparency since the bidding activity is transferred to automated agents. 4 Price-Based Iterative Combinatorial Auctions Many iterative CAs are price based and provide ask prices to guide bidding. In this section we survey some of these auction designs. We limit our attention to auctions designed for valuations that are rich enough to include the substitutes valuations. As such, we exclude the assignment model in which bidders have unit-demand for items. See Bikhchandani and Ostroy (Chapter 8) for a taxonomy that includes this case. 16

17 All the auctions that we discuss share the same high level structure: In each round the auctioneer announces ask prices and a provisional allocation and requests new bids from bidders. The bids are used to formulate a new winner-determination problem and update the provisional allocation, and also to adjust ask prices and test for termination. Table 2.1 provides a summary of the characteristics of some well-known auctions, stating properties for straightforward (non-strategic) bidding. For the cases in which an auction terminates with the VCG outcome this assumption is justified in an ex post equilibrium but otherwise one should expect incentives for demand reduction. The auctions are described in terms of the structure of the price space, the bidding language, and the method used to update prices. 17

18 Name Valuations Price structure Bidding Price Update Outcome language method KC substitutes non-anon items OR-items greedy CE SAA substitutes items OR-items greedy CE GS substitutes items XOR minimal min CE 12 Aus substitutes items single greedy a VCG ibundle; Ascending-proxy b BSM non-anon bundles XOR greedy VCG... general min CE dvsv BSM non-anon bundles XOR minimal VCG Clock-proxy BSM items (+ proxy) c XOR greedy VCG... general min CE RAD general items OR LP-based AkBA general anon bundles XOR LP-based ibea general non-anon bundles XOR greedy d VCG MP general non-anon bundles XOR minimal d VCG Table 2.1: Price-Based Combinatorial Auctions. Formal properties are stated for straightforward bidding, and with the most general class of valuations for which the property holds. Notation in the Outcome column indicates that no formal properties have been established. Notes: a Aus traces n + 1 trajectories. b Ascending-proxy dynamics are identical to ibundle(3), although ascending-proxy emphasizes a sealed-bid proxy auction form. c Clock-proxy is a hybrid design, with a linear-price clock auction followed by a sealed-bid ascending-proxy auction. d Ascending price while the auction is open, followed by a downwards adjustment after termination. Abbreviations: KC (Kelso and Crawford 1982) SAA (Milgrom 2000) GS (Gul and Stacchetti 2000) Aus (Ausubel 2002) ibundle (Parkes and Ungar 2000a) Ascending-proxy (Ausubel and Milgrom 2002) dvsv (de Vries, Schummer, and Vohra 2003) Clock-proxy (Ausubel and Milgrom, Chapter 5) RAD (Kwasnica et al. 2003) AkBA (Wurman and Wellman 2000) ibea (Parkes and Ungar 2002) MP (Mishra and Parkes 2004) We see a wide variety of prices, from simple prices on items (linear prices) to non-anonymous prices on bundles (non-anonymous and non-linear). In addition, the auctions vary in the bids that a bidder can submit in each round: OR-items, an additive-or bid for multiple items; XOR, an exclusive-or bid for multiple bundles; single, a bid on a single bundle in each round; OR, an additive-or bid for multiple 18

19 bundles. The XOR language has emerged as the definitive choice in recent designs. The price-update methods, which characterize the rules by which prices are computed in each round, are broken down as follows: Greedy update: The price is increased on some arbitrary set (perhaps all) of the over-demanded items or bundles. Minimal update: The price is increased on a minimal set of overdemanded items, or based on the bids from a set of minimally undersupplied bidders. LP-based: A linear program, formulated to find prices that are good approximations to CE prices given current bids, is used to adjust prices. For linear prices, Demange, Gale and Sotomayor (1986) in the assignment model and later Gul and Stacchetti (2000) for substitutes define a minimal update in terms of increasing the prices on a minimal overdemanded set of items. 13 Minimal price updates are adopted to drive prices towards minimal CE prices. de Vries, Schummer and Vohra (2003) generalize this to define updates in terms of minimally undersupplied bidders 14 and define a minimal update for general CAs. All bidders in a minimally undersupplied set face higher prices on the bundles for which they submitted a bid. RAD and AkBA adopt LP-based price updates and adjust prices to find good approximations to CE prices given current bids and the current provisional allocation. RAD seeks linear and anonymous prices while AkBA seeks non-linear but anonymous price approximations. Formal convergence properties have not been proved for RAD or AkBA, although RAD reduces to a simultaneous ascending price auction for substitutes valuations. The auctions that are able to implement the VCG outcome (for instance, Aus for substitutes and dvsv for BSM coalitional values) are interesting because they bring straightforward bidding into an equilibrium. Straightforward bidding is a 19

20 best response, whatever the valuations of other bidders, as long as the other bidders also follow a straightforward (perhaps untruthful) bidding strategy. This ex post equilibrium concept is useful because it places no requirements on the knowledge that bidders have of the valuations of other bidders. Winning bidders pay their final bid price in all auctions except Aus, ibea and MP. Aus allows for (n + 1) restarts and uses information elicited along each trajectory to determine the final payments. ibea and MP terminate with UCE prices, at which point final payments are determined through downwards adjustments. Auction clock-proxy (Ausubel, Cramton and Milgrom Chapter 5) is a hybrid auction. The first stage maintains item prices and runs an ascending-clock CA (see also Porter, Rassenti, and Smith (2003)). This stage is used for price discovery and can be considered to construct approximate linear CE prices. The second stage is sealed-bid, with bids from the first stage combined with additional bids that must be consistent with bids from the clock phase. 4.1 Insufficiency of Simple Prices It is interesting to consider what form of prices are necessary to implement efficient ascending CAs. Gul and Stacchetti (2000) first addressed this question, in the setting of substitutes valuations. The authors provide a formal definition of an ascending CA, but limit attention to linear and anonymous prices. They show that there exists no ascending VCG auction with linear and anonymous prices for substitutes valuations. The auction due to Ausubel (2002) lies outside of this negative characterization because it uses n + 1 price trajectories. Recently, Mishra and Parkes (2004) used the UCE-based price characterization to demonstrate that efficient ascending CAs require both non-anonymous and non-linear prices, even for this case of substitutes valuations. The authors exhibit instances for which only non-anonymous and non-linear UCE prices exist. As for sufficiency, auctions dvsv and ibundle are examples of ascending VCG auctions for substitutes valuations that maintain these rich prices. 20

21 However, de Vries, Schummer, and Vohra (2003) extend the definition of ascending CAs in Gul and Stacchetti (2000) to allow for non-anonymous and non-linear prices and obtain a negative result. When at least one bidder has a non-substitutes valuation an ascending CA cannot implement the VCG outcome even when the other bidders are restricted to substitutes and even with non-anonymous and non-linear prices. Auctions ibea and MP lie outside of this negative characterization because they allow a final downwards adjustment to determine final prices. Thus, with substitutes values but simple prices we must accept auctions with multiple trajectories or non-monotonic adjustments. Moreover, although rich prices extend the reach of ascending CAs to substitutes values we still need to accept multiple trajectories or non-monotonic adjustments to handle richer preferences than substitutes. 4.2 Primal-Dual Auction Design Many traditional combinatorial optimization problems can be solved with primal-dual algorithms. A primal-dual approach uses linear-programming (LP) duality to formulate an optimization problem as a satisfaction problem. Strong LP duality states that a pair of feasible primal and dual solutions are optimal if and only if they satisfy complementary slackness (CS) conditions. We provide a brief review of LP theory at the end of this chapter, and refer the reader to Papadimitriou and Steiglitz (1998) for a textbook treatment. In fact, primal-dual theory also provides a useful conceptual framework for the design of iterative price-based CAs. Prices are interpreted as a feasible dual solution and the provisional allocation is interpreted as a feasible primal solution. Bids provide sufficient information to formulate and solve restricted primal and dual problems, the winner-determination and price-update problems respectively (see Figure 2.2). For further discussion of this idea, see Parkes (2001), de Vries, Schummer and Vohra (2003) and Bikhchandani and Ostroy (Chapter 8). 21

22 Straightforward bidding is first assumed, and later justified by termination with VCG payments. The winner-determination problem uses information implicit in bids to compute a feasible solution that minimizes the violation of the CS conditions, and price updates adjust the dual solution towards an optimal dual solution. 15 CS conditions have an exact equivalence with conditions (1) and (2) required for CE prices, and are satisfied on termination of an auction. Constructively, primal-dual auction design requires the following steps: 1. Formulate an LP for the CAP that is integral, such that its optimal solution is a feasible allocation. The dual problem should allow convergence to UCE prices, or to minimal CE prices that support VCG payments in the case of BAS valuations. 2. Provide bidders with a bidding language that is expressive for straightforward bidding, and formulate a winner-determination problem to compute a feasible primal solution that minimizes the violation of CS conditions as represented in bids. 3. Terminate when the provisional allocation and ask prices satisfy CS conditions (and thus represent a CE), and also satisfy any additional conditions that are necessary to compute the VCG payments at termination (e.g. UCE conditions or minimal CE prices). Otherwise, adjust prices to make progress towards an optimal dual solution that satisfies these conditions. The characterization of VCG payments in terms of minimal CE and UCE prices suggests two methods to adjust towards the VCG outcome. The methods are illustrated in Figure 2.3, which considers the price on bundles, S 1 and S 2, allocated to bidders 1 and 2 in the efficient outcome. In case (a), the coalitional value function satisfies BSM and the VCG payments are supported at the minimal CE prices. Ascending CAs (such as dvsv) can 22

23 converge monotonically to these prices and the VCG outcome. In case (b), the coalitional value function satisfies neither BSM not BAS. Although each bidder s VCG payment is supported in some minimal CE there is no single CE that supports the VCG payment to both bidders simultaneously. As illustrated, ascending CAs such as ibea and MP can still converge monotonically to UCE prices from which the VCG outcome can be determined in a final adjustment. The next section presents a case study of primal-dual methods to the design and analysis of the ibundle auction. 16 In Section 4.4 we return to the auctions in Table 2.1, and discuss each in a little more detail. 4.3 Case Study: ibundle We will focus on variation ibundle(2), in which prices are non-linear but anonymous. This variation is efficient with straightforward bidding and an additional requirement that bidder strategies satisfy a safety property. Later, we also briefly describe ibundle(3), which employs non-linear and non-anonymous prices and is efficient without the safety condition. The interested reader is referred to Parkes and Ungar (2000a) and Parkes (2001) for additional details, including a description of ibundle(d), which blends ibundle(2) and ibundle(3) and allows for dynamic price discrimination decisions to be made during the auction. In what follows, we will use ibundle to refer to variation ibundle(2) unless otherwise stated. ibundle(2): Anonymous Prices ibundle maintains ask prices on bundles and a provisional allocation and proceeds in rounds, indexed t 1. In each round a bidder can submit XOR bids on bundles. In general the bid price on a bundle must be at least the ask price. Bidders must resubmit bids on any bundle that they are winning in the current provisional allocation but can bid at the same price on such a bundle even if the ask price has since increased. A bidder can also bid at ǫ less than the ask price 23

24 when making a last-and-final bid, at which point she can no longer improve her price. Equivalently, one can simply retain all bids from previous rounds. A bid at, or above, the current ask price is said to be competitive, and a bidder is competitive if she submits at least one competitive bid. The winner-determination problem in each round is to compute a provisional allocation to maximize the seller s revenue given bids, with at most one bundle selected from the XOR bid of each bidder. Let B i denote the bids from bidder i, and p bid,i (S) denote the bid price on bundle S B i. Winner determination can be formulated as the following mathematical program: s.t. max x i (S)p bid,i (S) x i (S) i I S B i x i (S) 1, i (5) S B i x i (S) 1, j (6) i I S B i :j S x i (S) {0,1}, i, S B i Constraint (5) restricts the seller to selecting at most one bid from each bidder. Constraint (6) ensures the allocation is feasible. Ties are broken first to favor the allocation from the previous round and then to maximize the number of winning bidders. ibundle terminates when each competitive bidder receives a bundle in the provisional allocation. Otherwise, prices are increased, by ǫ above the bid price on all bundles that receive a bid from some losing bidder in the current round and the new allocation and prices are provided as feedback to bidders. Prices on other bundles are implicitly adjusted to satisfy free disposal, although only bundles that receive losing bids need to be explicitly quoted. On termination the provisional allocation becomes the final allocation, and bidders pay their final bid prices. ibundle maintains feasible primal and dual solutions to an extended LP formulation of CAP and terminates with a CE outcome that satisfies CS 24

25 conditions. The proof technique is inspired by Bertsekas (1987) analysis of the AUCTION algorithm for the special case of unit-demand valuations. Given ask prices, p i (S), to bidder i we define ǫ straightforward bidding in terms of an ǫ-demand set, ǫ-ds, which is: ǫd i (p i ) = {S : v i (S) p i (S) + ǫ max S (v i (S ) p i (S ),0), S G} (ǫ-ds) In words, bidders state in their bid all bundles that come within ǫ of maximizing their surplus given prices in each round. This reduces to straightforward bidding for a small enough ǫ. Definition 5 (Safety). The competitive bundles in the ǫ-demand set of each losing bidder in each round are non-disjoint, i.e. each pair of bundles shares at least one item. For example, losing bids {(ABC, $100),(CDE, $50)} from a single bidder satisfy safety, while losing bids {(ABC, $100),(DE, $50)} from a single bidder fail the safety condition. Theorem 6. (Parkes and Ungar 2000a) ibundle(2) terminates with an allocation that is within 3min(n,m)ǫ of the efficient solution for ǫ-straightforward bidding strategies and with bid safety. The first step of the proof is to introduce an extended LP formulation (LP 2 ) for CAP due to Bikhchandani and Ostroy (2002, see also Chapter 8). LP 2 is integral when the safety condition holds for straightforward bidding. The dual formulation (DLP 2 ) has variables that correspond to anonymous and non-linear prices. Let K denote the set of feasible partitions. For example, (A, B, C) and (AB, C) are feasible partitions for items ABC. Variable y(k) = 1 will indicate that the allocation must be restricted to bundles in partition k K. For example, if partition (AB, C) is selected then the only valid allocations are those in which 25

26 AB goes to some bidder and C to another bidder. We have: s.t. max x i (S),y(k) S G i I i I x i (S)v i (S) [LP 2 ] x i (S) 1, i S G x i (S) y(k), S y(k) 1 k K k K:S k x i (S),y(k) 0, i,s,k min π i + Π s [DLP 2 ] π i,p(s),π s i I s.t. π i + p(s) v i (S), i,s Π s p(s) 0, k S k π i,p(s),π s 0, i,s Dual variable p(s) can be interpreted as the ask price on bundle S. Then, optimal πi = max S{v i (S) p(s),0} defines the maximal payoff to bidder i across all bundles given prices, and optimal Π s = max k K S k p(s) defines the maximal revenue to the seller across all partitions given prices. This is also the maximal revenue across all allocations because prices are anonymous. The dual problem sets prices to minimize the sum of the maximal payoff to each bidder and the maximal revenue to the seller. Optimal dual prices will correspond to CE prices whenever the primal LP is integral. Interpret the provisional allocation and ask prices in a round of ibundle(2) as defining a feasible primal and a feasible dual solution (denoted ˆx,ŷ, ˆπ i, ˆp, and ˆΠ s ). We can now establish termination with CS conditions for straightforward bidding strategies. The first primal CS condition is: ˆx i (S) >0 ˆπ i + ˆp(S) = v i (S), i,s (CS-1) 26

27 This states that any bundle allocated to bidder i must maximize her payoff across all bundles at the prices. Condition (CS-1) is approximately satisfied in every round because the provisional allocation is selected with respect to bids, which are in turn drawn from ǫ demand sets. Formally, a relaxed form of condition (CS-1) holds, with ˆx i (S) > 0 ˆπ i + ˆp(S) v i (S) + 2ǫ, for all i and S. The second primal CS condition is: ŷ(k) > 0 ˆΠ s ˆp(S) = 0, k (CS-2) S k This states that the provisional allocation must maximize the seller s payoff (i.e. revenue) given the prices, across all feasible allocations and irrespective of bids received from bidders. Bundle S has a strictly positive price if it is greater than the price on every bundle contained in S. Then, (CS-2) follows from properties (P1) and (P2), which are maintained in each round of the auction: (P1) All bundles with strict positive prices receive a bid from some bidder in every round. (P2) One or more of the revenue-maximizing allocations in every round can be constructed from bids from different bidders. Formally, (P1) follows because one can show that a losing bidder will continue to bid for S in the next round, even at the higher price. Property (P2) follows from the safety property, which prevents a single bidder from causing the price to increase on a pair of disjoint bundles. This is why we need the safety condition. Combining (P1) and (P2), and together with ǫ-ds, we get a relaxed formulation of (CS-2), with ŷ(k) > 0 ˆΠ s S k ˆp(S) min(m,n)ǫ, for all partitions k K. Dual CS condition (CS-3), states: ˆπ i > 0 ˆx i (S) = 1, i (CS-3) S G 27

28 In words, every bidder with positive payoff for some bundle at the current prices must receive a bundle in the provisional allocation. (CS-3) is satisfied for all bidders that receive bundles in a particular round, but not for the losing bidders that are still competitive. However, (CS-3) holds for every bidder on termination because at this point ǫ DS = for all losing bidders. (CS-3) and (CS-1) are equivalent to CE condition (1) and (CS-2) together with an additional requirement that a provisional allocation is always selected is equivalent to CE condition (2). Finally, we obtain an upper-bound on the worst-case efficiency error of ibundle, in terms of the minimal bid increment ǫ. First, sum the approximate (CS-1) condition over all bidders in the final allocation, and substitute ˆπ i = 0 for bidders not in the allocation by (CS-3). This gives: ˆπ i v i (Ŝi) ˆp(Ŝi) + 2min(m,n)ǫ (7) i I i I i I ˆΠ s + ˆπ i v i (Ŝi) + 3min(m,n)ǫ (8) i I i I where Eq. (7) follows because an allocation can include no more bundles than there are items or bidders, and Eq. (8) is by substitution of the ǫ-approximate (CS-2) condition. The LHS of Eq. (8) is the value of the final dual solution, and the first-term on the RHS is the value of the final primal solution. Now, ˆΠ s + i ˆπ i w(i), (the value of the optimal primal) by LP weak duality, and therefore w(i) ˆΠ s + i ˆπ i i v i(ŝi) + 3min(m,n)ǫ. A complete proof must also show termination. The basic idea is to assume the auction never terminates and prove that a bidder must eventually submit a bid at a price above her valuation, assuming finite values and a finite number of items, from which we get a contradiction with straightforward bidding. 28

29 ibundle(3): Non-anonymous Prices ibundle(3) is the variation of ibundle in which each bidder faces non-anonymous prices in every round. The dynamics of ibundle(3) with straightforward bidding are identical to that of Ausubel and Milgrom s (2002) ascending-proxy auction, although ascending-proxy is not described in price terms. ibundle(3) is efficient for straightforward bidding with general values. Moreover, the auction will terminate with VCG outcomes for BSM coalitional value functions. Let p t ask,i (S) denote the ask prices to bidder i in round t. Initially, p1 ask,i (S) = 0 for all bundles S and all bidders. Bids are received, and the winner determination problem solved, as in ibundle(2). Then, for each bidder not in the provisional allocation, the price to that bidder is increased by the minimal bid increment, ǫ > 0, above her bid price on all bundles submitted in that round, and adjusted for free-disposal. It is now quite immediate to establish that ibundle(3) terminates in CE with straightforward bidding. The prices faced by a bidder in round t are parameterized by πi t 0, which can be interpreted as the maximal payoff to the bidder in that round. The ask price on bundle S in round t is defined as: p t ask,i (S) = max(0,v i(s) π t i) (9) Initially, πi 1 = max S{v i (S)}, for all i, and the price is zero on all bundles. The payoff πi t decreases monotonically during the auction and prices monotonically increase. The ǫ-ds for bidder i in round t includes every bundle for which v i (S) πi t, and increases monotonically across rounds. Eventually, when πt i is less than ǫ the prices on each bundle are within ǫ of her value and she will bid for every bundle with positive value in her ǫ-ds. 17 Condition (CS-1) holds trivially in each round and (CS-3) holds at termination, just as in ibundle(2). In addition, (CS-2) holds in each round because of the special structure of prices: every bundle with a strict positive price receives a bid in a bidder s ǫ-ds. This does not require the safety condition. 29